Subgroup ($H$) information
| Description: | $C_3^4:S_3$ |
| Order: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(7,8,9)(10,11,12)(13,14,15), (1,3,2)(4,6,5)(10,15,12,14,11,13), (10,12,11)(13,14,15), (1,3,2), (7,14,11)(8,15,12)(9,13,10), (1,2,3)(4,5,6)(13,14,15)\rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $C_3^5:D_6$ |
| Order: | \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3\times (C_3\times \PSU(3,2)).S_3^3$ |
| $\operatorname{Aut}(H)$ | $C_3^3.S_3^3$ |
| $\operatorname{res}(S)$ | $C_3^2.S_3^3$, of order \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(9\)\(\medspace = 3^{2} \) |
| $W$ | $C_3^2:C_6$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_3^3$ | |||
| Normalizer: | $C_3^5:S_3$ | |||
| Normal closure: | $C_3^5:S_3$ | |||
| Core: | $C_3^3:S_3$ | |||
| Minimal over-subgroups: | $C_3^5:S_3$ | |||
| Maximal under-subgroups: | $C_3^4:C_3$ | $C_3^3:S_3$ | $S_3\times C_3^3$ | $C_3\wr S_3$ |
Other information
| Number of subgroups in this autjugacy class | $8$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $C_3^3:S_3^2$ |